A hydrogen atom changes its state from n=3 to n=2 . Due to recoil, the percentage change in the wave length of emitted light is approximately 1 10^ -n . The value of n is_____. [Given Rhc =13.6 eV , hc =1242 eV nm , h =6.6 10⁻³⁴ ~J ~s mass of the hydrogenatom =1.6 10⁻²⁷ ~kg ]

Step-by-step solution

aligned & E =13.6 ( 1 2^2 - 1 3^2 )=1.9 eV & E = hc & = hc E & P _ i = P _ t & 0=- mv + h ^ & v = h m ^ & E = 1 2 mv^ 2 + hc ^ & = 1 2 ~m ( h m ^ )^2+ hc ^ aligned Now aligned & E = h^2 2 ~m ^ ^2 + hc ^ & ^ 2 E - hc ^ - h ^2 2 ~m =0 & ^ = hc h ^2 c ^2+ 4 Eh ^2 2 ~m 2 E & ^ = hc hc 1+ 2 E mc 2 E aligned aligned & ^ = 1+ (1+ 2 E mc ^2 )^ 1 2 2 = 1+1+ E mc ^2 2 & ^ =1+ E 2 mc ^2 & ^ - = E 2 mc ^2 = 1.9 1.6 10⁻¹⁹ 2 1.67 10⁻²⁷ 9 10¹⁵ =10⁻⁹ & % change 10⁻⁷ aligned Correct answer 7